===== Zeta functions =====
==== Note ====
=== A bit on encodings and basic operations ===
== Field algebra ==

Let $S$ be the space/statespace/system 

(maybe with parts/states/aspects $s_0, s_1, s_2, s_3, s_4,\dots$)

== traceless part ==

$S \equiv 1 + T \equiv 1 - t$ 

or similar ... here $1$ is the neutral/constant thing in the theory and $T$ resp. $t$ is what's really interesting about $S$.

E.g. you have a scattering matrix and $1$ is the free propagation and $T$ is the interaction.

== Q ==

$\dfrac{1}{S}$ ... flipped encoding, switches low and far behavior, represents the weight of $X$.

$Q(t):=\dfrac{1}{1-t}=\sum_{n=0}^\infty t^n$ 

It starts out as $Q(t)=1+t+{\mathcal O}(t^2) \approx 1-T$, but it diverges once $t$ reaches $1$.

(And I observe $S\,Q(t)=2\,Q(t)-1$.)

== log ==

$\log(S)=\log(1+T)=\sum_{n=0}^\infty\frac{(-1)^n}{n}T^n$ 

... logarithmic encoding, alternating+declining coefficients give very good convergence.

We want to understand this, in a broad sense, as tamed version of the original: $\log(T) < T$. 

But, for $Q(t)$ interpreted in a field, its proper singularity isn't tamed by $\log$: 

$\log(Q(t))=\log\left(\dfrac{1}{1-t}\right)=-\log(1-t)=\sum_{n=0}^\infty\frac{1}{n}t^n$

still diverges at $\lim{t\to 1}$.

== zeta ==
$\zeta_S$ ... Some gluing together of data of $S$. 

Sometimes zetas are somewhat obscured using $\exp$'s chained with $\log$'s, in the spirit of above.

== Product ==
$\Pi$ is a gluing together of some aspects of a system. 

It's a convolution (in the literal and the metaphorical sense) of structure (e.g addition for polynomials and multiplication for Dirichlet-like objects such as the Riemann zeta function)

== Riemann zeta ==
For primes $p$, set $t_z=p^{-z}$ and define

$\zeta_\text{Riemann}(z):=\prod Q(t_z)=\prod_\text{primes p}\frac{1}{1-p^{-z}}$.

== Polylog ==
See [[Polylogarithm]]:

$\log\left(\dfrac{1}{1-t}\right)={\mathrm{Li}}_1(t)$

$\zeta_\text{Riemann}(z)={\mathrm{Li}}_s(1)$

=== Reference ===
Wikipedia: 
[[http://en.wikipedia.org/wiki/Local_zeta-function|Local zeta-function]],
[[http://en.wikipedia.org/wiki/Weil_conjectures#Statement_of_the_Weil_conjectures|Weil conjectures # Statement of the Weil conjectures]]

StackExchange:
[[http://math.stackexchange.com/questions/429616/what-is-a-zeta-function|What is a zeta function?]] (Great answer!)

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=== Related ===
[[Riemann zeta function]]